2.4 Use a General Strategy to Solve Linear Equations

Before you get started, take this readiness quiz.

Simplify: $− \left(a - 4\right) .$
If you missed this problem, review the reference.

Multiply: $\frac{3}{2} \left(12 x + 20\right)$.
If you missed this problem, review the reference.

Simplify: $5 - 2 \left(n + 1\right)$.
If you missed this problem, review the reference.

Multiply: $3 \left(7 y + 9\right)$.
If you missed this problem, review the reference.

Multiply: $\left(2.5\right) \left(6.4\right)$.
If you missed this problem, review the reference.

Solve: $5 \left(x + 3\right) = 35 .$

Solve: $6 \left(y - 4\right) = −18 .$

Solve: $− \left(y + 8\right) = −2 .$

Solve: $− \left(z + 4\right) = −12 .$

Solve: $2 \left(m - 4\right) + 3 = −1$.

Solve: $7 \left(n - 3\right) - 8 = −15$.

Solve: $\frac{1}{3} \left(6 u + 3\right) = 7 - u$.

Solve: $\frac{2}{3} \left(9 x - 12\right) = 8 + 2 x$.

Solve: $12 - 3 \left(4 j + 3\right) = −17$.

Solve: $−6 - 8 \left(k - 2\right) = −10$.

Solve: $6 \left(p - 3\right) - 7 = 5 \left(4 p + 3\right) - 12$.

Solve: $8 \left(q + 1\right) - 5 = 3 \left(2 q - 4\right) - 1$.

Solve: $6 \left[4 - 2 \left(7 y - 1\right)\right] = 8 \left(13 - 8 y\right)$.

Solve: $12 \left[1 - 5 \left(4 z - 1\right)\right] = 3 \left(24 + 11 z\right)$.

Solve: $0.55 \left(100 n + 8\right) = 0.6 \left(85 n + 14\right)$.

Solve: $0.15 \left(40 m - 120\right) = 0.5 \left(60 m + 12\right)$.

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:

$4 + 9 \left(3 x - 7\right) = −42 x - 13 + 23 \left(3 x - 2\right)$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:

$8 \left(1 - 3 x\right) + 15 \left(2 x + 7\right) = 2 \left(x + 50\right) + 4 \left(x + 3\right) + 1$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: $11 \left(q + 3\right) - 5 = 19$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: $6 + 14 \left(k - 8\right) = 95$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:

$12 c + 5 \left(5 + 3 c\right) = 3 \left(9 c - 4\right)$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:

$4 \left(7 d + 18\right) = 13 \left(3 d - 2\right) - 11 d$

Solve each linear equation.

$21 \left(y - 5\right) = −42$

Solve each linear equation.

$−16 \left(3 n + 4\right) = 32$

Solve each linear equation.

$5 \left(8 + 6 p\right) = 0$

Solve each linear equation.

$− \left(t - 19\right) = 28$

Solve each linear equation.

$8 \left(9 b - 4\right) - 12 = 100$

Solve each linear equation.

$21 + 2 \left(m - 4\right) = 25$

Solve each linear equation.

$−6 + 6 \left(5 - k\right) = 15$

Solve each linear equation.

$8 \left(6 t - 5\right) - 35 = −27$

Solve each linear equation.

$−2 \left(11 - 7 x\right) + 54 = 4$

Solve each linear equation.

$\frac{3}{5} \left(10 x - 5\right) = 27$

Solve each linear equation.

$\frac{1}{4} \left(20 d + 12\right) = d + 7$

Solve each linear equation.

$15 - \left(3 r + 8\right) = 28$

Solve each linear equation.

$−3 - \left(m - 1\right) = 13$

Solve each linear equation.

$18 - 2 \left(y - 3\right) = 32$

Solve each linear equation.

$35 - 5 \left(2 w + 8\right) = −10$

Solve each linear equation.

$−2 \left(a - 6\right) = 4 \left(a - 3\right)$

Solve each linear equation.

$5 \left(8 - r\right) = −2 \left(2 r - 16\right)$

Solve each linear equation.

$9 \left(2 m - 3\right) - 8 = 4 m + 7$

Solve each linear equation.

$−15 + 4 \left(2 - 5 y\right) = −7 \left(y - 4\right) + 4$

Solve each linear equation.

$5 \left(x - 4\right) - 4 x = 14$

Solve each linear equation.

$−12 + 8 \left(x - 5\right) = −4 + 3 \left(5 x - 2\right)$

Solve each linear equation.

$7 \left(2 n - 5\right) = 8 \left(4 n - 1\right) - 9$

Solve each linear equation.

$3 \left(a - 2\right) - \left(a + 6\right) = 4 \left(a - 1\right)$

Solve each linear equation.

$− \left(7 m + 4\right) - \left(2 m - 5\right)$
$= 14 - \left(5 m - 3\right)$

Solve each linear equation.

$5 \left[9 - 2 \left(6 d - 1\right)\right]$
$= 11 \left(4 - 10 d\right) - 139$

Solve each linear equation.

$3 \left[−14 + 2 \left(15 k - 6\right)\right]$
$= 8 \left(3 - 5 k\right) - 24$

Solve each linear equation.

$10 \left[5 \left(n + 1\right) + 4 \left(n - 1\right)\right]$
$= 11 \left[\right. 7 \left(5 + n\right) - \left(25 - 3 n\right) \left]\right.$

Solve each linear equation.

$4 \left(2.5 v - 0.6\right) = 7.6$

Solve each linear equation.

$0.2 \left(p - 6\right) = 0.4 \left(p + 14\right)$

Solve each linear equation.

$0.5 \left(16 m + 34\right) = −15$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$15 y + 32 = 2 \left(10 y - 7\right) - 5 y + 46$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$9 \left(a - 4\right) + 3 \left(2 a + 5\right) = 7 \left(3 a - 4\right) - 6 a + 7$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$24 \left(3 d - 4\right) + 100 = 52$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$30 \left(2 n - 1\right) = 5 \left(10 n + 8\right)$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$18 u - 51 = 9 \left(4 u + 5\right) - 6 \left(3 u - 10\right)$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$5 \left(p + 4\right) + 8 \left(2 p - 1\right) = 9 \left(3 p - 5\right) - 6 \left(p - 2\right)$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$9 \left(4 k - 7\right) = 11 \left(3 k + 1\right) + 4$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$60 \left(2 x - 1\right) = 15 \left(8 x + 5\right)$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$36 \left(4 m + 5\right) = 12 \left(12 m + 15\right)$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$11 \left(8 c + 5\right) - 8 c = 2 \left(40 c + 25\right) + 5$

Coins Rhonda has $1.90 in nickels and dimes. The number of dimes is one less than twice the number of nickels. Find the number of nickels, n, by solving the equation $0.05 n + 0.10 \left(2 n - 1\right) = 1.90$.

Explain why you should simplify both sides of an equation as much as possible before collecting the variable terms to one side and the constant terms to the other side.

Solve the equation $\frac{1}{4} \left(8 x + 20\right) = 3 x - 4$ explaining all the steps of your solution as in the examples in this section.